Mehler's Hermite polynomial formula is the bilinear generating function
![sum_(n=0)^infty(H_n(x)H_n(y))/(n!)(1/2w)^n=(1-w^2)^(-1/2)exp[(2xyw-(x^2+y^2)w^2)/(1-w^2)],](/images/equations/MehlersHermitePolynomialFormula/NumberedEquation1.svg) |
(1)
|
where 
is a Hermite polynomial and 
(Watson 1933; Erdélyi 1938; Szegö 1975,
p. 380). The generating function
 |
(2)
|
where 
is the floor function, can be derived from this
equation (Doetsch 1930; Szegö 1975, p. 380). The more straightforward sum with 
replaced by 
in the denominator is given by
 |
(3)
|
A generalization with shifted Hermite polynomial indices uses the function 
defined by either side of Mehler's formula. For nonnegative integers 
and 
, real numbers 
and 
,
and 
, put 
, 
, and 
. Then
 |  |  |
(4)
|
Related generalizations are discussed by Tapia-Valerdi et al. (2024). The identity follows by comparing coefficients
of 
in
![e^(2xu-u^2+2yv-v^2)S(x-u,y-v;w)=S(x,y;w)exp[(2(x-wy)u+2(y-wx)v-u^2-v^2+2wuv)/h],](/images/equations/MehlersHermitePolynomialFormula/NumberedEquation4.svg) |
(5)
|
using the exponential generating function of the Hermite polynomials. The case 
recovers Mehler's formula.
